A note on estimation of functionals from indirect observations with or without sparsity

The objective of the paper is to develop a theory of estimation of functionals based on indirect observations when the underlying vector of unobservable parameters is possibly sparse. The paper uses a novel approach of reduction of the problem of estimation of functionals in indirect observations $\Phi_n = n^{-1} \sum_{i=1}^n \phi (\theta_i)$ to estimation of linear functionals $\Phi = \int_{-\infty}^\infty \phi(x) f(x) dx$ of an unknown deconvolution density $f$. This allows to research a variety of problems as one paradigm and obtain answers to a multitude of questions. The paper offers a general, Fourier transform based approach to estimation of $\Phi$ (and $\Phi_n$) and provides the upper and the minimax lower bounds for the risk in the case when function $\phi$ is square integrable, and the vector of unobservable parameters is sparse or not. The theory is extended to some situations when $\phi$ is not square integrable. As a direct result of application of the proposed theory, we recover and extend multiple recent results such as estimation of the mixing cumulative distribution function, estimation of the first absolute moment based on indirect observations and estimation of the mixing probability density function with classical and Berkson errors.